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By Adem A., Milgram R.J., Ravenel D.C. (eds.)

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If Q is any cube, the surface area and volume of Q are related by vol(∂Q) = 2n vol(Q)(n−1)/n . ˜ for which We wish to show that any other convex body K has an affine image K ˜ ≤ 2n vol(K) ˜ (n−1)/n . vol(∂ K) ˜ so that its maximal volume ellipsoid is B2n , the Euclidean ball of Choose K ˜ is then at most 2n , since this is the volume of the radius 1. The volume of K cube whose maximal ellipsoid is B2n . As in the previous lecture, n ˜ ˜ ˜ = lim vol(K + εB2 ) − vol(K) . vol(∂ K) ε→0 ε ˜ ⊃ B n , the second expression is at most Since K 2 n ˜ + εK) ˜ − vol(K) ˜ vol(K ˜ lim (1 + ε) − 1 = vol(K) ε→0 ε→0 ε ε ˜ (n−1)/n ˜ ˜ 1/n vol(K) = n vol(K) = n vol(K) ˜ (n−1)/n , ≤ 2n vol(K) lim which is exactly what we wanted.

Let C be the maximum possible ratio x /|x| for nonzero x and let θ be a point of S k−1 with θ = C. Choose ψ in the δ-net with |θ − ψ| ≤ δ. Then θ − ψ ≤ C|θ − ψ| ≤ Cδ, so C = θ ≤ ψ + θ − ψ ≤ (1 + γ) + Cδ. Hence (1 + γ) . 1−δ To get the lower bound, pick some θ in the sphere and some ψ in the δ-net with |ψ − θ| ≤ δ. Then C≤ (1 − γ) ≤ ψ ≤ θ + ψ − θ ≤ θ + (1 + γ)δ (1 + γ) |ψ − θ| ≤ θ + . 1−δ 1−δ Hence θ ≥ 1−γ− δ(1 + γ) 1−δ = (1 − γ − 2δ) . 1−δ  According to the lemma, our approach will give us a slice that is within distance 1+γ 1 − γ − 2δ of the Euclidean ball (provided we satisfy the hypotheses), and this distance can be made as close as we wish to 1 if γ and δ are small enough.

Milman, “New volume ratio properties for convex symmetric bodies in Rn ”, Invent. Math. 88 (1987), 319–340. [Bourgain et al. 1989] J. Bourgain, J. Lindenstrauss, and V. Milman, “Approximation of zonoids by zonotopes”, Acta Math. 162 (1989), 73–141. [Brascamp and Lieb 1976a] H. J. Brascamp and E. H. Lieb, “Best constants in Young’s inequality, its converse and its generalization to more than three functions”, Advances in Math. 20 (1976), 151–173. [Brascamp and Lieb 1976b] H. J. Brascamp and E. H.

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